Algorithm 5 – Lattice
2.1 Fuzzy sets
2 Fuzzy Sets and Fuzzy Connectives
Fuzzy sets extend classical set theory in order to deal with uncertainty. Whereas classical mathematics works on the assumption that we have absolute precision concerning data (we know whether or not an element belongs to a given set, or we know exactly how a function behaves within some neighbourhood of a point), fuzzy logic deals with imprecisions.
There are of course social conventions which behave classically, such as the age from which someone is allowed to drive, or to drink alcoholic beverages. Nonetheless, considerations on how such legislations vary across different countries indicates that these conventions do not correspond precisely to reality, i.e., in reality people are sufficiently mature to drink or drive at different ages, but because maturity is a subjective matter and the legislation should (in theory at least) treat individuals objectively, it establishes a number with some degree of arbitrariness1.
In the present chapter we present the required theory of fuzzy sets and fuzzy connectives as it shall be necessary in the following chapters.
which is often called the universe of discourse2.
Recall that subsets of U may be represented by their characteristic functions, i.e., for each APP♣Uq we have χA P2Y given by
χA♣xq ✏
✩✫
✪
1, if xPA 0, if x❘A .
Fuzzy sets are defined by extending 2✏ t0,1✉to I ✏ r0,1s ❸R, thus enabling us to work with uncertain degrees of membership, i.e., rather than stating something or its negation (1 and 0, respectively), we are allowed to say something in between.
Definition 2.1.1. Let U be a (classical) set. A fuzzy subset F of U is a function3 µF :U ÑI .
The set of all fuzzy subsets ofU is denoted byIU.
Henceforth we shall frequently refer to fuzzy subsets as fuzzy sets, as long as there is no risk of confusion concerning the set U. It is a common practice to state that F is characterized by the function µF, which is called the membership function of F. In order to unify this practice and our definition, henceforth we shall consider a fuzzy set and its membership function to be the same entity, and at times we may say that a fuzzy set is characterized by its membership function. We may also denote the set of fuzzy subsets of U byF♣Uq4.
A fuzzy set µsuch that µ♣uq P t0,1✉ for every u P U is called a crisp set. If the set U ✏ tu1, ..., un✉ is finite and A is a fuzzy subset of U we may write
A✏ µA♣u1q u1
... µA♣unq un
.
Example 2.1.2. What does it mean to be young? Individuals a and b, aged 20 and 80 respectively, may have very different opinions about individual c, aged 40, being young.
2 By speaking about a "universe of discourse" we do not state that there is a "universal set", i. e., a set which contains all sets as elements of itself as one such set would be subject to Russell‘s paradox (HRBÁČEK; JECH,1999) . On the contrary, the universe of discourse is a set conceived in order to avoid paradoxes.
3 Some clarification may be useful here. Many authors (such as (ZADEH, 1965) and (BARROS;
BASSANEZI; LODWICK,2017)) defineF as beingcharacterized by a functionµF. We do understand that this is a good idea in terms of intuition, and we shall use this idea later for practical purposes.
Nonetheless, this idea presents a philosophical difficulty: if F ischaracterized byµF, then whatis F itself? We then decide to define a fuzzy set to bethe functionµF. Additionally, this definition is in accordance with the definition ofL-fuzzy sets stated later in this chapter.
4 This is merely a notation. Although for classical sets there is a distinction between 2U andP♣Uqand they are related by a bijection, in the fuzzy caseIU andF♣Uqare one and the same set.
Nonetheless they are all likely to agree that Mr. a is young, whereas Mr. b is not. Having this in mind, we may define µY, the membership to the set of young people, as follows5:
Let U ✏ tn PN:n↕120✉. Define6
µY♣xq ✏
✩✬
✬✬
✫
✬✬
✬✪
1, if 0↕x↕20 60✁x
40 , if 20➔x↕60 0, if 60➔x↕120 .
0 20 40 60 80 100 120
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4 – Membership to the Fuzzy Set of Young People
Given two classical sets A, B ❸U, their intersection (A❳B) and union (A❨B) are expressed by the following characteristic functions:
χA❳B♣xq ✏
✩✫
✪
1, if xPA and xP B
0, if x❘A or x❘B (or both) , χA❨B♣xq ✏
✩✫
✪
0, if x❘A and x❘ B
1, if xPA or xPB (or both) .
The complement of A (that is, the set U③A) denoted byA✶, has the following characteristic function:
5 We are aware that age alone is no sufficient criteria for defining someone as young or old. As sang by the children in the Mexican TV showEl Chavo del Ocho, "There are young people in their eighties and there are old people who are 16 years old." Nonetheless we do choose to work only with age for simplicity.
6 Notice that the points atx✏20 andx✏60 give us constraints. If we want to make the membership function (when extended tor0,120s) of classCn, each of these points give us two additional constraints, meaning that it could be achieved by fitting a 2n✁1 degree polynomial to the 2nconstraints. However, high degree polynomials constrained to a limited interval may become wavy. If we used a function of the form 1④1 e♣r sxq we would have a smooth function that approaches but never reaches 0 nor 1.
χA✶♣xq ✏
✩✫
✪
0, if xPA 1, if x❘A . Notice that for any xPU we have the following7:
χA❳B♣xq ✏ χA♣xq ❫χB♣xq , χA❨B♣xq ✏ χA♣xq ❴χB♣xq ,
χA✶♣xq ✏ 1✁χA♣xq .
Now one sees that extending the classical definitions is an easy matter. The definitions presented below are called the standard fuzzy set operations.
Definition 2.1.3. Let A, B be fuzzy subsets of U. Then the fuzzy subsets A❳B and A❨B, called the intersection and union of A and B respectively, and the set A✶, called the complement of A, have the following membership functions:
µA❳B♣xq ✏ µA♣xq ❫µB♣xq , µA❨B♣xq ✏ µA♣xq ❴µB♣xq ,
µA✶♣xq ✏ 1✁µA♣xq .
When working with fuzzy sets, properties distinct from those of classical sets often arise.
Example 2.1.4. From the setY of young people defined in Example2.1.2, we shall define the set O of old people as the complement Y✶ of Y (Fig. 5a), that is:
µO♣xq ✏1✁µY♣xq ✏
✩✬
✬✬
✫
✬✬
✬✪
0, if 0↕x↕20 x✁20
40 , if 20➔x↕60 1, if 60➔x↕120
Thus, we get the following results for Y ❨O and Y ❳O (Fig. 5b):
µY❨O♣xq ✏
✩✬
✬✬
✬✬
✬✫
✬✬
✬✬
✬✬
✪
1, if 0↕x↕20 60✁x
40 , if 20➔x↕40 x✁20
40 , if 40➔x↕60 1, if 60➔x↕120
µY❳O♣xq ✏
✩✬
✬✬
✬✬
✬✫
✬✬
✬✬
✬✬
✪
1, if 0↕x↕20 60✁x
40 , if 20➔x↕40 x✁20
40 , if 40➔x↕60 1, if 60➔x↕120
7 Recall that ❫and ❴are the meet (infimum) and join (supremum), defined in Sec.0.2, on the lattice 2✏ t0,1✉with the usual order of natural numbers.
0 20 40 60 80 100 120 0.0
0.2 0.4 0.6 0.8 1.0
(a) Fuzzy SetOof Old People
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(b)Y ❨O (top) andY ❳O(bottom)
Figure 5 – Operations with Fuzzy Subsets
Notice that differently from the classical case, the union and intersection of a fuzzy set with its complement may not be the full set U or the empty set. For example, µY❨O♣40q ✏0.5✏µY❳O♣40q.
As we have seen in Sec. 0.28, given A, B ❸U, A❸B iff χA ↕χB ,
whereχA ↕χB expressesχA♣uq ↕χB♣uqfor alluPU. We extend this criteria of subsetness to fuzzy sets.
Definition 2.1.5. Let A, B be fuzzy sets of U. We say that A is a fuzzy subset of B, denoted by A❸B, iff
µA ↕µB , that is, iff for all xPU, µA♣xq ↕ µB♣xq9.
Considering that we want to have, for every fuzzy subset A of U, the relations
❍ ❸ A❸U, it follows that the membership functions of U and ❍ are pointwise given by µU♣xq ✏1 and µ❍♣xq ✏ 0, respectively.